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I. Basic math.
II. Pricing and Hedging.
1. Basics of derivative pricing I.
A. Single step binary tree argument. Risk neutral probability. Delta hedging.
B. Why Ito process?
C. Existence of risk neutral measure via Girsanov's theorem.
D. Self-financing strategy.
E. Existence of risk neutral measure via backward Kolmogorov's equation. Delta hedging.
F. Optimal utility function based interpretation of delta hedging.
2. Change of numeraire.
3. Basics of derivative pricing II.
4. Market model.
5. Currency Exchange.
6. Credit risk.
7. Incomplete markets.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Single step binary tree argument. Risk neutral probability. Delta hedging.

onsider a market consisting of a single stock and a money market account (MMA). The MMA is riskless and has rate $r$ . The price of the stock is given by a stochastic process $S_{t}$ . At present moment $t_{0}$ the stock is priced at $S_{t_{0}}=S_{0}$ . There is no trading or price evolution until a time moment $t_{1}$ , $t_{1}>t_{0}$ and at $t_{1}$ the stock may assume only two possible prices: $S_{1}$ and $S_{2}$ . Hence, we introduce the random events MATH We would like to come up with a $t_{0}$ -price for a contract that pays at time $t_{1}$ a random amount MATH

Suppose that at the moment $t_{0}$ we sell the contract for a price $h_{0}$ , purchase $x$ shares of the stock and put the remainder of the funds $y$ into the MMA. Any of the values MATH may be negative. We assume that we can short the stock and borrow from MMA at the same rate $r$ .

The value of the position MATH is zero: MATH We hold the position until the time moment $t_{1}$ when there are two possibilities: MATH Let us select $x$ and $y$ so that MATH would be the same for both situations: MATH MATH We arrive to a system MATH of two equations: MATH The quantity MATH is called "delta" and the strategy of taking position in the underlying stock equal to MATH is called "delta hedging".

To determine $y$ we make the following "no arbitrage" argument. At $t_{0}$ we have a position of zero value. At the moment $t_{1}$ we have a position of set value (given our selection of MATH ). Such set value has to be zero because otherwise we made or lost money in absence of risk, hence, the price $h_{0}$ of the contract would not be correct. Therefore, we have a third equation: MATH from which we derive $y$ : MATH and we obtain the correct price of the contract: MATH We would like to put the above result into the form MATH for some numbers $q_{1},q_{2}$ . Hence, we rearrange the expression $\left( \&\right) $ : MATH thus MATH Note that MATH For this reason we call the numbers $q_{1},q_{2}$ the "risk neutral probabilities": MATH and represent the expression MATH as the "risk neutral expectation of discounted payoff": MATH

Notation. Index. Contents.

Copyright 2007